Metamath Proof Explorer


Theorem hstpyth

Description: Pythagorean property of a Hilbert-space-valued state for orthogonal vectors A and B . (Contributed by NM, 26-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion hstpyth ( ( ( 𝑆 ∈ CHStates ∧ 𝐴C ) ∧ ( 𝐵C𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ( norm ‘ ( 𝑆 ‘ ( 𝐴 𝐵 ) ) ) ↑ 2 ) = ( ( ( norm ‘ ( 𝑆𝐴 ) ) ↑ 2 ) + ( ( norm ‘ ( 𝑆𝐵 ) ) ↑ 2 ) ) )

Proof

Step Hyp Ref Expression
1 hstosum ( ( ( 𝑆 ∈ CHStates ∧ 𝐴C ) ∧ ( 𝐵C𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( 𝑆 ‘ ( 𝐴 𝐵 ) ) = ( ( 𝑆𝐴 ) + ( 𝑆𝐵 ) ) )
2 1 fveq2d ( ( ( 𝑆 ∈ CHStates ∧ 𝐴C ) ∧ ( 𝐵C𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( norm ‘ ( 𝑆 ‘ ( 𝐴 𝐵 ) ) ) = ( norm ‘ ( ( 𝑆𝐴 ) + ( 𝑆𝐵 ) ) ) )
3 2 oveq1d ( ( ( 𝑆 ∈ CHStates ∧ 𝐴C ) ∧ ( 𝐵C𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ( norm ‘ ( 𝑆 ‘ ( 𝐴 𝐵 ) ) ) ↑ 2 ) = ( ( norm ‘ ( ( 𝑆𝐴 ) + ( 𝑆𝐵 ) ) ) ↑ 2 ) )
4 hstcl ( ( 𝑆 ∈ CHStates ∧ 𝐴C ) → ( 𝑆𝐴 ) ∈ ℋ )
5 4 adantr ( ( ( 𝑆 ∈ CHStates ∧ 𝐴C ) ∧ ( 𝐵C𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( 𝑆𝐴 ) ∈ ℋ )
6 hstcl ( ( 𝑆 ∈ CHStates ∧ 𝐵C ) → ( 𝑆𝐵 ) ∈ ℋ )
7 6 ad2ant2r ( ( ( 𝑆 ∈ CHStates ∧ 𝐴C ) ∧ ( 𝐵C𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( 𝑆𝐵 ) ∈ ℋ )
8 hstorth ( ( ( 𝑆 ∈ CHStates ∧ 𝐴C ) ∧ ( 𝐵C𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ( 𝑆𝐴 ) ·ih ( 𝑆𝐵 ) ) = 0 )
9 normpyth ( ( ( 𝑆𝐴 ) ∈ ℋ ∧ ( 𝑆𝐵 ) ∈ ℋ ) → ( ( ( 𝑆𝐴 ) ·ih ( 𝑆𝐵 ) ) = 0 → ( ( norm ‘ ( ( 𝑆𝐴 ) + ( 𝑆𝐵 ) ) ) ↑ 2 ) = ( ( ( norm ‘ ( 𝑆𝐴 ) ) ↑ 2 ) + ( ( norm ‘ ( 𝑆𝐵 ) ) ↑ 2 ) ) ) )
10 9 3impia ( ( ( 𝑆𝐴 ) ∈ ℋ ∧ ( 𝑆𝐵 ) ∈ ℋ ∧ ( ( 𝑆𝐴 ) ·ih ( 𝑆𝐵 ) ) = 0 ) → ( ( norm ‘ ( ( 𝑆𝐴 ) + ( 𝑆𝐵 ) ) ) ↑ 2 ) = ( ( ( norm ‘ ( 𝑆𝐴 ) ) ↑ 2 ) + ( ( norm ‘ ( 𝑆𝐵 ) ) ↑ 2 ) ) )
11 5 7 8 10 syl3anc ( ( ( 𝑆 ∈ CHStates ∧ 𝐴C ) ∧ ( 𝐵C𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ( norm ‘ ( ( 𝑆𝐴 ) + ( 𝑆𝐵 ) ) ) ↑ 2 ) = ( ( ( norm ‘ ( 𝑆𝐴 ) ) ↑ 2 ) + ( ( norm ‘ ( 𝑆𝐵 ) ) ↑ 2 ) ) )
12 3 11 eqtrd ( ( ( 𝑆 ∈ CHStates ∧ 𝐴C ) ∧ ( 𝐵C𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ( norm ‘ ( 𝑆 ‘ ( 𝐴 𝐵 ) ) ) ↑ 2 ) = ( ( ( norm ‘ ( 𝑆𝐴 ) ) ↑ 2 ) + ( ( norm ‘ ( 𝑆𝐵 ) ) ↑ 2 ) ) )